Growing Patterns in One Dimensional Cellular Automata

We st udy limi t evolut ions of cellular automata (CA) both theoretically and exp erimentally. We show that eit her all orbits ente r the limit set after less than a finit ely bo unded number of states, or almost all orbits never ent er the limit set . We link this result with a class ification of CAs according to their limit behavior due to Culik et al.: in t he first case, the considered CA belongs to class 1 while in the second case, it belongs to class 2. By experiments, we try to measure the convergenc e spee d of orbits to their acc umulation points. We compute t he maximum number of nest ed growing segments in a space-time diagram representing a finit e portion of an orbit. We observe that , in the average case , this criterion dep ends only on the CA and not on the configur at ion . We also observe . two kinds of CA wit h resp ect to our crite rion which correspond to the intuiti ve notions of chaos and regulari ty. We do fur ther experiments to explain t he fact t hat the proportion of chaotic diagrams grows with the number of states of t he considered CA .